In  the second author proposed to find a description (or examples) of real-valued -variable functions satisfying the following two inequalities:
with strict inequality if there is an index such that ; and for , then, In this short note we extend in a direction a result of  and we prove a theorem that provides a large class of examples satisfying the two inequalities, with replaced by any symmetric polynomial with positive coefficients. Moreover, we find that the inequalities are not specific to expressions of the form , rather they hold for any function that satisfies some conditions. A simple consequence of this result is a theorem of Hardy, Littlewood and Polya .
|if , then , || |
 G. HARDY, J.E. LITTLEWOOD and G. PÓLYA, Inequalities, Cambridge Univ. Press, 2001.
 P. STANICA, Inequalities on linear functions and circular
powers, J. Ineq. in Pure and Applied Math., 3(3) (2002),